Suppose $\Omega \subset \mathbb{R}^n$ is Lipshitz domain, $u \in L^q(\Omega)$.
I am reading some article that assume as clear that
$$\frac{\partial u}{\partial x_i} \in W^{-1,q}(\Omega).$$
Is that true?
I can show that $\frac{\partial u}{\partial x_i} \in W^{-1,q'}(\Omega)$ since for $\phi \in W^{1,q'}(\Omega)$ i have by Green's formula
$$\int_\Omega \frac{\partial u}{\partial x_i} \phi\, dx = \int_{\partial \Omega} u \phi \nu_i \, dS - \int_\Omega u \frac{\partial \phi}{\partial x_i} \, dx$$
and since both integrals on the right hand side make sense I can identify $\frac{\partial u}{\partial x_i}$ with that particular linear functional and thus $\frac{\partial u}{\partial x_i} \in W^{-1,q'}(\Omega)$. But is also $\frac{\partial u}{\partial x_i} \in W^{-1,q}(\Omega)$?